DRRAiNN's structure is grounded in the following data and structural information sources. The locations Li=(xi,yi) for estimations of discharge in the river network are determined by discharge gauging stations that provide observed discharge Qi,t for time t in 24h periods. The connectivity of stations, determined by the river network, is encoded in an adjacency matrix Ai,j. Static maps Sx,y and meteorological forcings Fx,y,t for hourly time points t are encoded on a grid that spans a domain larger than the elevation-delineated catchment area to capture effective catchment contributions beyond topographic boundaries. Given static maps S:,:, meteorological forcings F:,:,t0:ts+T over the whole duration (t0…ts+T) in hours, and past discharge Qi,t0:ts over the tune-in period (t0…ts) in days, DRRAiNN estimates discharge Qi,ts+1:ts+T over a temporal future horizon of T days via a function f, representing the learned spatio-temporal mapping implemented by the model: 1Q̃i,ts+1:ts+T=f(S:,:,F:,:,t0:ts+T,Qi,t0:ts) Since surface and subsurface flow differ from river flow dynamics as described above, we model these subprocesses separately. Therefore, DRRAiNN consists of two components, the rainfall-runoff model and the discharge model. The rainfall-runoff model operates recurrently on a grid, rendering it fully distributed. It is supposed to model surface and subsurface flow, and evapotranspiration. The discharge model operates recurrently on a graph to model river flow inside of channels and output estimated discharge Q̃ at the station locations. While DRRAiNN is fully distributed in its internal computation over a spatial grid, its outputs are only available at selected gauging stations.

At each time step, DRRAiNN processes the sequence in an auto-regressive loop by first invoking the rainfall-runoff model, followed by the discharge model. The rainfall-runoff model receives gridded static maps S and meteorological forcings F as input to model the catchment on a grid. It is primed to distinguish between two important subprocesses, namely surface and subsurface flow, which is mainly driven by topography, and evapotranspiration, which is mainly driven by solar radiation. It produces a latent representation, which we term runoff embedding, extracted at station locations and used as input to the discharge model. Despite being the main driver of discharge, it cannot be directly interpreted as runoff due to its self-organizing nature. The discharge model additionally receives an adjacency matrix A that describes the connectivity between stations, static river segment features, and the (potentially estimated) discharge Q:,t-1 from the previous time step. It then estimates discharge Q̃ for each station, from which the training loss is computed.

We implement DRRAiNN in pytorch . In the following, we provide a more detailed description of DRRAiNN's components. See Fig.  for a depiction of the overall model.

The input data for DRRAiNN consists of radar-based precipitation, elevation for above-ground topography, solar radiation, and river discharge data. Preliminary experiments showed no improvement when including temperature; therefore, we exclude it following Occam's razor.

For precipitation, we use the radar-based precipitation product RADOLAN provided by the Deutsche Wetterdienst . The data domain is a 900km×900km pixel grid with a resolution of 1km×1km that covers all of Germany and a temporal resolution of 1h. This grid defines the spatial resolution at which our model operates. RADOLAN data is log-standardized before being sent to the model due to its long-tail distribution. Specifically, we add 1 and take the logarithm, then compute the mean and standard deviation of the transformed data to standardize it. We replace missing values with zeros, which is the standardized mean.

For static topography information we use the digital elevation model (DEM) EU-DEM v1.1 provided by the Copernicus Land Monitoring Service of the European Environment Agency . We also use the DEM to compute the differences in altitudes between adjacent discharge gauging stations. Elevation values and derived difference are standardized before being sent to the model, i.e., we subtract their mean and divide by their standard deviation.

For solar radiation, we use surface short-wave downward radiation (SSRD) from the ERA5 data set . It comes with a temporal resolution of 1h and a relatively coarse spatial resolution of 0.25°×0.25°. Like the precipitation data, solar radiation data is log-standardized. We use rasterio to transform and reproject the DEM and solar radiation data to match the RADOLAN coordinate reference system.

The topography of our river network is determined by the AWGN data set . We use it to compute the adjacency matrix that describes which stations are connected via river segments and the corresponding river segment lengths.

Finally, we use discharge measurement data to tune in the discharge model and, more importantly, as the only target variable to train, validate, and test our model. We use data collected and provided by the German Federal Institute of Hydrology via the Global Runoff Data Centre . The data set contains observed daily river discharge from gauging stations worldwide, including those in Germany. Since the location information of the discharge gauging stations is partially wrong, we corrected them manually. We then align the station locations to the nearest river segment (snapping). If the correction exceeds a predefined threshold of 1km, the station is excluded. If two stations are very close to each other, one of them is discarded. Due to its long-tail distribution, discharge data is log-standardized on a per-station basis before being sent to the model. We add 1 and take the logarithm, then standardize the data using station-wise means and standard deviations. We replace missing values with zeros, which is the standardized mean of the corresponding station.

Our choice of input datasets was guided by temporal resolution, data provenance, and practical availability. Although the European Flood Awareness System (EFAS) employs EMO-1 for precipitation input, we opted for RADOLAN due to important differences: EMO-1 offers a coarser 6h resolution and is interpolated from sparse station data, in contrast to RADOLAN's direct radar-based observations. Although we expect only minor differences in performance in some settings, radar-derived datasets like RADOLAN provide finer spatial and temporal resolution, which is advantageous for distributed models. Similarly, we chose ERA5 for solar radiation data due to its gridded format and hourly resolution. Alternative datasets, such as those provided by DWD, are either available only as station-wise hourly data, which lack the required grid format, or as gridded data aggregated monthly, which does not meet our temporal requirements. Daily datasets like EOBS may suffice if subdaily temporal patterns are encoded separately, but this would require additional preprocessing. A transition toward operation flood forecast would place increased importance on the choice of precipitation forecast products . Ultimately, all data products entail inherent uncertainties and errors, and our choices reflect a balance between data availability, temporal resolution, and the specific requirements of our model.

The Neckar river network in Southwest Germany spans a catchment area of 14000km2 with a mean elevation of 460m. According to ERA5, mean temperatures in this region were 0.95°C during winter and 17.95°C during summer in our training period. Our dataset includes measurements from 17 gauging stations distributed across the river network (see Fig. ). At the most downstream station in Rockenau, discharge during the training period ranged from 29.5 to 1690m3s-1 with a mean of 133.3m3s-1.

The catchment features a highly heterogeneous landscape, including narrow and wide valleys, diverse geology (e.g., limestone, sandstone), different soil textures (e.g., clay, marl), and subsurface structures such as karst systems and pore water aquifers. This makes the modeling of the Neckar River network a challenging endeavor. To give a concrete example, there are underground flows south of Pforzheim that route water toward the east, while the elevation model suggests a different flow direction . This relationship cannot be inferred from a digital elevation model alone. Latent underground structures route the water in a different direction than the elevation model alone would suggest.

By restricting the domain to the Neckar river network, we end up with an area of size 200km×200km. Following the transformations described above, all gridded data is reduced from a 1km×1km grid to a 4km×4km grid by taking the mean. This results in a 50×50 grid covering the study area. We train our model on hydrological years 2006–2015, validate on 2016–2018, and test on 2019. Forcings F are provided at hourly resolution, while discharge is provided at daily resolution.

We train DRRAiNN on sequences of 20 d (480 h steps), using the first 10 d as a warm-up phase. During this phase, we feed the model observed discharge values to initialize and align its hidden states with the true system dynamics. This procedure resembles data assimilation in traditional hydrological models, where observations are used to update model states and reduce uncertainty. In ML terms, this corresponds to teacher forcing. The warm-up phase allows the rainfall-runoff component of DRRAiNN to infer latent hydrological states, such as soil moisture or aquifer recharge, through its hidden state representations. This alignment helps the model transition smoothly to predictive, open-loop mode, where future discharge is estimated without access to ground-truth values.

After the warm-up phase, DRRAiNN transitions into open-loop mode for the remaining 10 d of each sequence. In this predictive mode, the discharge model feeds its own previous discharge estimations as inputs for subsequent time steps. The rainfall-runoff model, in contrast, continues to receive observed precipitation and solar radiation as inputs throughout the sequence. While informative, this setup does not reflect realistic operational conditions for discharge forecasting. Precipitation forecasting, in particular, remains a major challenge. Currently no algorithm can accurately predict precipitation 10 d ahead at a spatial resolution of 4km×4km. However, this setup is well suited for knowledge discovery concerning hydrologic processes, which is primary focus in this work. We leave the evaluation of DRRAiNN under realistic, forecast-based conditions for future work.

We use the mean squared error (MSE) computed on station-wise standardized discharge data as both the training and validation loss. Standardization ensures that stations with larger discharge values do not dominate the loss, promoting a balanced learning across all stations. Training is performed using truncated backpropagation through time (TBPTT), where the truncation length increases progressively over the course of training. Initially, we backpropagate the loss over 1 d sequences (24 time steps) to help DRRAiNN focus on short-term temporal relationships and stabilize learning. Over the course of training, we increase the truncation length, enabling the model to learn longer-term dependencies. The truncation length schedule is shown in Table . Training takes less than 8h on a single NVIDIA A100 GPU. A forward simulation of a 20 d sequence takes approximately 4s.

To improve generalization and account for model variability due to random initialization, we train five independent instances of DRRAiNN per experiment, each initialized with a different seed. We report test results based on the three runs with the lowest validation loss out of five seeds. This selection procedure is applied consistently to both the primary model and all ablation variants. We use the Ranger optimizer with a learning rate of 0.0025 to optimize the 30600 parameters in DRRAiNN. To stabilize training, we clip the gradient if its norm exceeds 1, thereby preventing large parameter updates in steep regions of the loss surface. We use hydra to manage experiment configurations .

To increase the size of the training data set and improve generalization, we apply data augmentation. CNNs are not inherently invariant to rotations and reflections, meaning a model trained on the original data orientation may perform poorly when presented with rotated or reflected versions of the same physical scenario. The symmetry group of the square contains eight elements: the identity, rotations by 90, 180, and 270°, and reflection in the x, y, and both diagonal axes. For each training sequence, we apply a uniformly sampled symmetry to the spatial variables in each time step. We ensure physical consistency by tapping into the runoff embeddings at the transformed station locations. The river discharge model's graph structure remains unchanged by this augmentation, as graph neural networks are inherently invariant to such spatial transformations.

To provide context for DRRAiNN's performance, we compare it to the European Flood Awareness System (EFAS), an established and operational distributed process-based model. We use publicly available EFAS reanalysis data, which eliminates the need to tune EFAS ourselves. This avoids potential biases that could arise from allocating unequal tuning effort to the benchmark model versus our own model. While DRRAiNN achieves higher performance than EFAS in many scenarios, our primary aim is to demonstrate the potential of distributed neural networks for river discharge estimation, rather than merely outperforming EFAS.

EFAS simulates runoff on an approximately 1.5km×1.5km grid with a temporal resolution of 6h. It receives as inputs static maps describing topography, river networks, soil, and vegetation, as well as meteorological forcings such as precipitation, temperature, and potential evaporation.

While EFAS serves as a useful benchmark, the comparison to DRRAiNN is not perfectly fair due to fundamental differences in the input and output variables. Both models receive gridded meteorological forcings, but DRRAiNN additionally receives discharge measurements during the tune-in period. In contrast, EFAS does not use discharge measurements as input but relies on them for offline model calibration. Furthermore, DRRAiNN produces discharge estimates only at gauging station locations, whereas EFAS generates discharge predictions across the entire spatial grid. EFAS also relies on additional input variables not used by DRRAiNN, such as soil type, vegetation, temperature, and potential evapotranspiration. While this makes EFAS a powerful tool, it also limits its applicability in regions lacking such detailed input data. Another difference lies in the precipitation data used: EFAS relies on EMO-1, a 6h product interpolated from weather station data, whereas DRRAiNN uses RADOLAN, a radar-based dataset offering higher spatial and temporal resolution. As a result, a direct comparison between EFAS and DRRAiNN is not valid. Nonetheless, EFAS serves as a baseline to contextualize the expected performance range of DRRAiNN. We thus emphasize that our goal is not to directly compare performance but to provide a baseline that allows us to place the principled quality of DRRAiNN's performance with respect to alternative state-of-the-art forecasting approaches.

Besides visualizing hydrographs for selected gauging stations, we evaluate DRRAiNN using four standard metrics in hydrology: Kling-Gupta efficiency (KGE, ), Nash-Sutcliffe efficiency (NSE, ), Pearson's correlation coefficient (PCC), and the mean absolute error (MAE). We report all four metrics because each highlights different aspects of model performance, and no single metric is free from limitations . MAE is particularly intuitive, as it is expressed in the same unit as discharge and directly quantifies the average deviation between predictions and observations. However, because it lacks normalization, stations with larger discharge magnitudes contribute disproportionately to the overall MAE. PCC quantifies the strength of linear association between the observed and estimated discharges. While it captures shared variability, it is insensitive to systematic differences in scale or bias. To also capture the scale, the NSE was developed, which can be seen as a mean squared error that is weighted by the variance of the observed discharge. While the NSE does incorporate bias, it does so in normalized form scaled by the standard deviation of the target variable, which can mask individual contributions of different error components. The KGE was therefore developed to independently evaluate correlation, bias, and variability as separate components. When computing KGE and NSE values, we use station-wise means and variances calculated from the training data set, following the approach in . For KGE, NSE, and PCC, higher values indicate better performance, with a maximum of 1 representing a perfect match. In contrast, lower values of MAE are better, with 0 indicating a perfect fit.

During open-loop inference, we evaluate metrics separately for each open-loop step, where the first step resembles closed-loop estimation. This allows us to assess how model performance degrades with increasing lead times. Although DRRAiNN was only trained on sequences that span 20 d, we evaluate it on 50 d sequences to investigate its ability to generalize beyond the training horizon. Additionally, we will plot the performance of the models against the mean discharge of the different stations to identify potential systematic dependencies between flow magnitude and model accuracy. In all cases, we exclude the initial 10 d tune-in period before calculating metrics and producing plots.

With knowledge discovery being the main motivation of this work, we also test DRRAiNN for physical plausibility. A physically implausible model might learn spurious relationships in the data. It could, for example, exploit the DEM to encode local biases that lead to gains or losses of water not driven by meteorological forcings. By retrospectively inferring catchment areas from observed dynamics, we assess whether the rainfall-runoff model successfully propagates water across the landscape. The procedure is as follows: After a forward pass, we compute saliency maps by taking the gradient of the final discharge estimate with respect to the precipitation inputs. These maps tell us to which extent the model's output depends on the precipitation in each grid cell and time step. We multiply this gradient by the precipitation itself to focus the analysis on cells in which precipitation occurred. To examine how the attributions change over time, we split the sequence into subsequences of 5 d, over which we take the mean. We do this for each station separately and visualize the resulting attributions to identify which areas contribute most to discharge estimation at each station. To reduce noise, we repeat this process across all test sequences and average the resulting attribution maps.

We compare the resulting attributions with catchment areas delineated from elevation data using standard hydrological techniques, which are widely used in the field. To evaluate their agreement quantitatively, we employ the following measure when comparing DRRAiNN to the ablated models: For each station, the attributions are standardized to lie between 0 and 1 using min-max scaling. We then compute the Wasserstein distance between the attributions values inside the delineated catchment area and those outside it. A higher Wasserstein distance indicates greater differentiation between attributions inside and outside elevation-delineated catchment areas. This provides one measure of how well the model's learned patterns correspond to topographically-defined flow boundaries, though deviations may reflect either model limitations or genuine subsurface flow processes not captured by surface topography alone. This quantitative measure complements the qualitative comparison, providing stronger evidence for our model's ability to propagate water across the landscape in a physically plausible way. Specifically, it indicates that the model has implicitly learned the topographic structure of flow direction – i.e., that water generally flows downhill – solely from observed discharge dynamics.

EFAS produces hydrographs that match both the shape and magnitude of observed discharge, rendering it a strong contestant (Fig. ). As EFAS produces gridded outputs, it is necessary to extract outputs from EFAS grid cells that correspond to the station locations in order to make meaningful comparisons.

DRRAiNN also produces plausible hydrographs that closely match the observed discharges. This includes both low flows (Fig. a) and high flows (Fig. d). Since DRRAiNN operates autoregressively – using its own discharge estimates as input in the next time step – error can accumulate over time, leading to gradual decline in accuracy. Nonetheless, it is notable that the model is in general able to hit peaks even after almost 50 d, despite being trained only on 20 d sequences.

Overall, DRRAiNN outperforms EFAS in all considered metrics (Fig. ). Since EFAS does not incorporate discharge values during inference, we report its mean performance over lead times as constant. As described above, DRRAiNN's autoregressive nature causes errors to accumulate over time, leading to a gradual decline in performance at longer lead times.

The KGE plot (Fig. a) indicates that DRRAiNN is able to maintain strong performance over time. Averaged over the seeds, starting with a KGE of about 0.71, our model's estimations stay above those of EFAS during the entire estimation horizon of 50 d, despite having been trained only on 20 d sequences. In contrast, the NSE plot (Fig. b) shows gradual decline in performance over time with a decrease from 0.72 to 0.62 over the estimation horizon. Regardless, even after 50 d, all seeds show higher NSE values than EFAS. The PCC plot (Fig. c) shows a strong linear relationship between observed and estimated discharges, with an average value of about 0.9 at the start. DRRAiNN captures this relationship better than EFAS over the entire estimation horizon. Note that the linear correlation is also part of KGE and NSE. As the MAE allows direct interpretation, its plot (Fig. d) shows that EFAS is off by about 6.5m3s-1 on average, while DRRAiNN with 3.9m3s-1 on average on the first day produces a considerable smaller error. After about 25 d, EFAS yields a lower MAE on average.

All metrics reveal differences in performance across the model instances trained with different random seeds. However, the relative ranking of model instances varies depending on the specific metric and lead time. Some seeds perform better during the initial days, while others are better with greater lead times: For example, in the KGE plot (Fig. a), the ranking changes after about 42 d. The difference between instances are due to random weight initialization and the order of batches only. These stochastic factors may lead some instances to start the training with a larger bias towards capturing short-term, while others start with a larger bias towards capturing long-term relationships in the data.

The plots in Fig.  show that some stations consistently yield more accurate discharge estimates than others. This observation holds across all evaluation metrics. Which stations are harder to estimate, however, is different across the metrics, reflecting the distinct sensitivities each metric has, as discussed previously. Interestingly, both the different DRRAiNN instance and EFAS show partial agreement on which stations are more difficult to model. For example, the KGE values in Fig. a show that Altensteig and Stein are consistently easier to estimate, while Oppenweiler, Bad Imnau, and Murr are among the most challenging. The reasons for this discrepancy – such as differences in data quality, catchment size, land cover, or upstream complexity – could be analyzed in future work.

The regression lines indicate whether model performance correlates with average discharge levels across stations. We performed linear regression; the regression lines appear exponential due to the logarithmic scaling of the x-axis. All metrics, except MAE, show that both models tend to perform better at stations with higher discharges. This effect is more pronounced in EFAS, while our model exhibits a more balanced behavior. Both DRRAiNN and EFAS produce significantly larger MAEs with increased mean discharge (Fig. d). This is expected since MAE does not account for the stations' mean discharges or their variability in discharge, unlike the other metrics.

We observe that DRRAiNN implicitly infers catchment areas that align with topographically expected ones, as shown in Fig. . Darker areas indicate regions with higher importance of precipitation for estimating discharge at the corresponding station. These attribution patterns spatially overlap with the catchment areas delineated from elevation alone (depicted in red). The first four columns visualize attributions for subsequences of 5 d length to illustrate temporal changes in spatial influence. There is a tendency of the area of influence to increases in size the further we look into the past. This suggests that DRRAiNN propagates encoded water quantities along the landscape in a manner that aligns, at least to some extent, with physical flow processes. The last column shows attributions averaged over the whole 20 d sequences.

In the case of Pforzheim, DRRAiNN assigns low importance to an area in the lower right part, despite its inclusion in the delineated catchment area. This discrepancy could be related to known underground flows near Pforzheim, as reported in . In the absence of subsurface flows, water would be expected to pass through Pforzheim; however, due to the presence of underground flow paths, it instead moves towards the southeast, entering the Neckar River network via an alternative route. Our results suggests that DRRAiNN may have detected these unobservable underground flows from precipitation and discharge dynamics. However, this hypothesis arguably needs more investigation in the future.

Note that these results primarily serve as a proof of principle: We present results from the seed producing the clearest attributions; others yielded qualitatively worse results. However, it is important to keep in mind that DRRAiNN is trained on daily discharge measurements. Learning sharp catchment delineations would require the training data set to contain sequences in which it rained within the area, but not outside of it, over the extent of a 24h period. As precipitation is very dynamic on this time scale, the chances for this are relatively low. In the future, we expect sharper results if we go from daily to hourly discharge data.

To assess both the contributions of specific architectural components and the model's use of topographic information, we conducted a series of ablations on DRRAiNN (Appendix ). First, we examined whether DRRAiNN utilizes the DEM primarily for encoding flow physics or as positional information by training, validating, and testing it on a rotated DEM. This resulted in slightly worse performance and reduced alignment between attribution maps and elevation-delineated catchments (Appendix ). Next, we evaluated the model's inductive bias in distinguishing between spatially extended and local processes (Appendix ). Last, we removed the hypernetworks to examine their impact (Appendix ). Both ablations led to performance degradation across most metrics and lead times. However, the differences were not always significant. Notably, both ablated models show reduced correspondence between their attribution patterns and elevation-delineated catchment boundaries. The attribution maps either exhibit less coherent spatial structures or show systematic deviations from topographically-expected patterns, suggesting that these architectural components contribute to the model's ability to learn topographically-consistent flow representations.